A quadratic function has a squared input and graphs as a parabola, a curved hill or valley. For profit, “at least zero” means the curve must be on or above the horizontal axis. Graph the model in Desmos, click the zero-profit points, then read the stretch that qualifies. Don’t stop at one zero: the curve can dip below again.
Hints
- Hint 1
Translate “at least ” into an inequality, a comparison: . The words at least include the case where the company breaks even, so don’t leave out equality.
- Hint 2
An x-intercept is a point where the graph’s output is . Graph the profit model in Desmos and click both places where it meets the horizontal axis. What production levels do they show?
- Hint 3
The negative coefficient of the squared term makes this parabola a hill. Check whether the curve lies above or below the axis between its two zero-profit points. Do the points themselves count?
Step-by-step
Graph the profit and read the interval
Step 1Translate the profit condition
“At least profit” means . This inequality asks for every production level with positive profit or zero profit. The output is in thousands of dollars, but zero thousand dollars still means breaking even.
- Step 2
Find the zero-profit points
Type , using for and for profit so Desmos can graph the given model. Click both x-intercepts, where the output is . Desmos labels them and , so the company breaks even at and .
- Step 3
Read where profit is not negative
Between those intercepts, the curve is above the axis; outside them, it’s below. “At least” includes the zero-profit endpoints, so . The company makes at least zero weekly profit when it produces and sells from to hundred gadgets, inclusive. Choice C.